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Surgery and duality
Surgery, as developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall
and others is a method for comparing homotopy types of topological spaces with
diffeomorphism or homeomorphism types of manifolds of dimension >= 5. In this
paper, a modification of this theory is presented, where instead of fixing a
homotopy type one considers a weaker information. Roughly speaking, one
compares n-dimensional compact manifolds with topological spaces whose
k-skeletons are fixed, where k is at least [n/2]. A particularly attractive
example which illustrates the concept is given by complete intersections. By
the Lefschetz hyperplane theorem, a complete intersection of complex dimension
n has the same n-skeleton as CP^n and one can use the modified theory to obtain
information about their diffeomorphism type although the homotopy
classification is not known. The theory reduces this classification result to
the determination of complete intersections in a certain bordism group. The
restrictions are: If d = d_1 ... d_r is the total degree of a complete
intersection X^n_{d_1,..., d_r} of complex dimension n, then the assumption is,
that for all primes p with p(p-1) <= n+1, the total degree d is divisible by
p^{[(2n+1)/(2p-1)]+1}.
Theorem A. Two complete intersections X^n_{d_1,.,d_r} and X^n_{d'_1,\ldots ,
d'_s} of complex dimension n>2 fulfilling the assumption above for the total
degree are diffeomorphic if and only if the total degrees, the Pontrjagin
classes and the Euler characteristics agree.Comment: 48 pages, published versio
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